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<span class="kw">let </span><font color="Maroon" title="c1">TM</font> be   <a href="pcomps_1.html#V3" title="PCOMPS_1:attr.3">metrizable</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">I</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><br/>  for <font color="Olive" title="b2">A</font>, <font color="Olive" title="b3">B</font> being   <a href="topdim_1.html#NV2" title="TOPDIM_1:NV.2">finite-ind</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font>  st <font color="Olive" title="b2">A</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  &amp; <font color="Maroon" title="c1">TM</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b3">B</font></span>)</span> is  <a href="waybel23.html#V5" title="WAYBEL23:attr.5">second-countable</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <font color="Olive" title="b2">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">I</font> &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <font color="Olive" title="b3">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">I</font> holds <br/>(  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b3">B</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">I</font> &amp; <font color="Olive" title="b2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b3">B</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">I</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="topdim_1.html#NV2" title="TOPDIM_1:NV.2">finite-ind</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font>  st <font color="Olive" title="b1">A</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  &amp; <font color="Maroon" title="c1">TM</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">B</font></span>)</span> is  <a href="waybel23.html#V5" title="WAYBEL23:attr.5">second-countable</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <font color="Olive" title="b1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">I</font> &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <font color="Olive" title="b2">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">I</font> holds <br/>(  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Olive" title="b1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">B</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font> be   <a href="topdim_1.html#NV2" title="TOPDIM_1:NV.2">finite-ind</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">A</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  &amp; <font color="Maroon" title="c1">TM</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">B</font></span>)</span> is  <a href="waybel23.html#V5" title="WAYBEL23:attr.5">second-countable</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <font color="Maroon" title="c3">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">I</font> &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <font color="Maroon" title="c4">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">I</font> implies (  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">B</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:11"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
<font color="Maroon" title="c3">A</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">and </span><br/><a NAME="E2:11"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<font color="Maroon" title="c1">TM</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">B</font></span>)</span> is  <a href="waybel23.html#V5" title="WAYBEL23:attr.5">second-countable</a> 
 <span class="kw">and </span><br/><a NAME="E3:11"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
 <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <font color="Maroon" title="c3">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">I</font>
 <span class="kw">and </span><br/><a NAME="E4:11"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
 <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <font color="Maroon" title="c4">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">B</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  )</span><br/>

<a NAME="E5:11"/>
 <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topdim_1.html#T5" title="TOPDIM_1:th.5">TOPDIM_1:5</a></span>;<br/>
<a NAME="E6:11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A5</font></span>: 
 <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:11"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:11_1"/>
( <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">B</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  not <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">B</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1"><span class="kw">suppose </span></a><a NAME="E1:11_1_1"/>
<font color="Maroon" title="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">B</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">B</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:11_1_1"/>
(  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">B</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:11"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="ref" href="topdim_1.html#T6" title="TOPDIM_1:th.6">TOPDIM_1:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_2"><span class="kw">suppose </span></a><a NAME="E1:11_1_2"/><span class="lab"><font color="Green" title="E14">A6</font></span>: 
 not <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">B</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">B</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  )</span><br/><div class="add"><a NAME="E2:11_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A7</font></span>: 
 not <font color="Maroon" title="c1">TM</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 ;<br/><a NAME="E3:11_1_2"/>
(  not <font color="Maroon" title="c3">A</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  not <font color="Maroon" title="c4">B</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_1_2"><span class="lab"><font color="Green" title="E14">A6</font></span></a></span>;<br/><a NAME="E4:11_1_2"/><span class="kw">then </span>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <font color="Maroon" title="c3">A</font> or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <font color="Maroon" title="c4">B</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:11_1_2"><span class="lab"><font color="Green" title="E15">A7</font></span></a></span>;<br/><a NAME="E5:11_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A8</font></span>: 
<font color="Maroon" title="c2">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:11"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E4:11"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="int_1.html#T3" title="INT_1:th.3">INT_1:3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">AB</font> = <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">B</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> ;<br/><span class="kw">set </span><font color="Maroon" title="c6">Tab</font> = <font color="Maroon" title="c1">TM</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c5">AB</font>;<br/><a NAME="E7:11_1_2"/><span class="lab"><font color="Green" title="E17">A9</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">TM</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c5">AB</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">AB</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c8">b</font> = <font color="Maroon" title="c4">B</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">TM</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c5">AB</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/><a NAME="E9:11_1_2"/>
<font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">TM</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c5">AB</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/><a NAME="E10:11_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A10</font></span>: 
<font color="Maroon" title="c7">a</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="tsp_1.html#D2" title="TSP_1:def.2">TSP_1:def 2</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">F</font> being   <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a>   <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">TM</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c5">AB</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E12:11_1_2"/><span class="lab"><font color="Green" title="E19">A11</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c9">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topgen_4.html#D6" title="TOPGEN_4:def.6">TOPGEN_4:def 6</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">a</font> = <font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c11">b</font> = <font color="Maroon" title="c8">b</font> as   <a href="topdim_1.html#NV2" title="TOPDIM_1:NV.2">finite-ind</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">TM</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c5">AB</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="topdim_1.html#T21" title="TOPDIM_1:th.21">TOPDIM_1:21</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">AA</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">TM</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c5">AB</font></span>)</span> ;<br/> <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">AA</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">F</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c12">AA</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c9">F</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T78" title="ZFMISC_1:th.78">ZFMISC_1:78</a></span>
<br/>.= 
<font color="Maroon" title="c10">a</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:11_1_2"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="ref" href="zfmisc_1.html#T25" title="ZFMISC_1:th.25">ZFMISC_1:25</a></span>
<br/>.= 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">TM</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c5">AB</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T2" title="PRE_TOPC:th.2">PRE_TOPC:2</a></span>
;<br/><a NAME="E16:11_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A12</font></span>: 
<font color="Maroon" title="c12">AA</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">F</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">TM</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c5">AB</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#D11" title="SETFAM_1:def.11">SETFAM_1:def 11</a></span>;<br/><a NAME="E17:11_1_2"/>
<font color="Maroon" title="c12">AA</font> is  <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">D</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">TM</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c5">AB</font></span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tops_2.html#D2" title="TOPS_2:def.2">TOPS_2:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c13">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">AA</font> or <font color="Maroon" title="c13">D</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:11_1_2_2"/>
<font color="Maroon" title="c13">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">AA</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">D</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> </span><br/>

<span class="kw">hence </span><a NAME="E2:11_1_2_2"/>
<font color="Maroon" title="c13">D</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:11_1_2"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E18:11_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A13</font></span>: 
<font color="Maroon" title="c12">AA</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">F</font> is  <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tops_2.html#T16" title="TOPS_2:th.16">TOPS_2:16</a></span>;<br/><a NAME="E19:11_1_2"/>
 for <font color="Olive" title="b1">D</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">TM</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c5">AB</font></span>)</span>  st <font color="Olive" title="b1">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">AA</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">F</font> holds <br/>( <font color="Olive" title="b1">D</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <font color="Olive" title="b1">D</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">I</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">D</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">TM</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c5">AB</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">AA</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">F</font> implies ( <font color="Maroon" title="c13">D</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <font color="Maroon" title="c13">D</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">I</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:11_1_2_3"/><span class="lab"><font color="Green" title="E22">A14</font></span>: 
<font color="Maroon" title="c13">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">AA</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">D</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <font color="Maroon" title="c13">D</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">I</font> )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_2_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:11_1_2_3_1"/>
( <font color="Maroon" title="c13">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">AA</font> or <font color="Maroon" title="c13">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">F</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_1_2_3"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_2_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:11_1_2_3_1_1"/>
<font color="Maroon" title="c13">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">AA</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">D</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <font color="Maroon" title="c13">D</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">I</font> )</span><br/><div class="add"><a NAME="E2:11_1_2_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:11_1_2_3_1_1"/>
( <font color="Maroon" title="c13">D</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <font color="Maroon" title="c13">D</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">I</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:11"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="topdim_1.html#T21" title="TOPDIM_1:th.21">TOPDIM_1:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_2_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:11_1_2_3_1_2"/><span class="lab"><font color="Green" title="E23">A15</font></span>: 
<font color="Maroon" title="c13">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">D</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <font color="Maroon" title="c13">D</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">I</font> )</span><br/><div class="add"><a NAME="E2:11_1_2_3_1_2"/>
<font color="Maroon" title="c10">a</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:11_1_2"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="ref" href="xboole_1.html#T40" title="XBOOLE_1:th.40">XBOOLE_1:40</a></span>;<br/><a NAME="E3:11_1_2_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A16</font></span>: 
<font color="Maroon" title="c10">a</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/><a NAME="E4:11_1_2_3_1_2"/>
<font color="Maroon" title="c13">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">a</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:11_1_2"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E1:11_1_2_3_1_2"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="ref" href="zfmisc_1.html#T74" title="ZFMISC_1:th.74">ZFMISC_1:74</a></span>;<br/><a NAME="E5:11_1_2_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A17</font></span>: 
<font color="Maroon" title="c13">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:11_1_2_3_1_2"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E6:11_1_2_3_1_2"/><span class="kw">then </span>
(  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <font color="Maroon" title="c11">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <font color="Maroon" title="c4">B</font> &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <font color="Maroon" title="c13">D</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <font color="Maroon" title="c11">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topdim_1.html#T19" title="TOPDIM_1:th.19">TOPDIM_1:19</a>, <a class="ref" href="topdim_1.html#T21" title="TOPDIM_1:th.21">TOPDIM_1:21</a></span>;<br/><span class="kw">hence </span><a NAME="E7:11_1_2_3_1_2"/>
( <font color="Maroon" title="c13">D</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <font color="Maroon" title="c13">D</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">I</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:11"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E5:11_1_2_3_1_2"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="ref" href="topdim_1.html#T19" title="TOPDIM_1:th.19">TOPDIM_1:19</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E20:11_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A18</font></span>: 
( <font color="Maroon" title="c12">AA</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">F</font> is  <a href="topdim_1.html#V2" title="TOPDIM_1:attr.2">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">AA</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">I</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:11"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="ref" href="topdim_1.html#T11" title="TOPDIM_1:th.11">TOPDIM_1:11</a></span>;<br/><a NAME="E21:11_1_2"/><span class="lab"><font color="Green" title="E23">A19</font></span>: 
<font color="Maroon" title="c12">AA</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">F</font> is  <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T85" title="CARD_2:th.85">CARD_2:85</a></span>;<br/><a NAME="E22:11_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">TM</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c5">AB</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:11"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E5:11_1_2"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E16:11_1_2"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E18:11_1_2"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E20:11_1_2"><span class="lab"><font color="Green" title="E22">A18</font></span></a>, <a class="txt" href="topdim_2.html#E8"><span class="lab"><font color="Green" title="E7">Lm3</font></span></a></span>;<br/><a NAME="E23:11_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A20</font></span>: 
<font color="Maroon" title="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">B</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topdim_1.html#T18" title="TOPDIM_1:th.18">TOPDIM_1:18</a></span>;<br/><a NAME="E24:11_1_2"/>
 <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">TM</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c5">AB</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:11"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E5:11_1_2"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E16:11_1_2"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E18:11_1_2"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E20:11_1_2"><span class="lab"><font color="Green" title="E22">A18</font></span></a>, <a class="txt" href="#E21:11_1_2"><span class="lab"><font color="Green" title="E23">A19</font></span></a>, <a class="txt" href="topdim_2.html#E8"><span class="lab"><font color="Green" title="E7">Lm3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E25:11_1_2"/>
(  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">B</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:11_1_2"><span class="lab"><font color="Green" title="E24">A20</font></span></a>, <a class="ref" href="topdim_1.html#T17" title="TOPDIM_1:th.17">TOPDIM_1:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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